I'm having a lot of trouble with an algebra question, or rather, with my lack of logic in understanding how it should be solved. I'm just going to quote the whole bit and then try to explain how I'm thinking it through by showing how I'm solving it.

The problem is:

Jared places a small inheritance of $2475 in a certificate of deposit that earns 6% interest compounded quarterly. The total in the CD after 10 years is given by the expression

2475 (1+0.06/4)^4(10).

I can already tell you the answer is 4489.70, according to the back of the book and online algebra calculators and I understand how to solve it, so that's fine. My problem is that I don't understand how the algebra problem fits with the word problem.

If I had looked at the word problem without looking at the expression, I would have assumed that what I needed to do is take 6% of 2475, which is $148.50, multiply that by 4 to represent the quarterly interest, which would give me $594 and then multiply that number by 10 to represent the 10 years that the CD is collecting interest, which is $5940. Then, to actually show the total in the CD I'd add the original $2475 to $5940.

So I guess what I'm saying is that if I were to come up with an expression on my own for the word problem, it would look more like

2475(0.06 times 40) + 2475

(I'm not too familiar with algebraic expressions, so I don't know if I did that right.)

I would just like someone to explain to me in exactly what way I'm wrong. I know I am wrong, but it's just that my way seems logical to me even though it can't be.

FrancesKatherine wrote:I would have assumed that what I needed to do is take 6% of 2475, which is $148.50, multiply that by 4 to represent the quarterly interest, which would give me $594 and then multiply that number by 10 to represent the 10 years that the CD is collecting interest, which is $5940. Then, to actually show the total in the CD I'd add the original $2475 to $5940.

Consider a simpler situation of $100 at 6% annual interest, compounded quarterly. By your method, one would compute the simple interest for the year, which would be (0.06)($100) = $6. Then one would multiply this by the four quarters to get $24 in interest.

But of course interest does not work with way. (Else, one's credit-card balances and mortgage payments would be staggeringly larger.) Instead, one does the computations at every compounding, and one does not award the full year's interest amount at every compounding. (This is why "effective" rates are always very close to the listed rates. The compoundings increase the amounts paid or owed, but only by a little over any one year's time.)

If interest is compounded twice a year, then one does the computations twice a year. Then the interest rate (let's continue to use 6%) is awarded for only half the year. To account for this halving, one uses half the interest rate: ($100)(0.06/2) = ($100)(0.03) = $3. Then, for the second half of the year, one awards the 3% interest on the new amount: ($103)(0.03) = $3.09. So the value at the end of the year will be $100 + $3 + $3.09 = $106.09.